The exceptional simple-factor conjecture for profinite groups with positive proportion of 2-elements

Let $\Sigma$ be the infinite family of finite non-abelian simple groups

$$\operatorname{PSL}(2,3^f),\qquad f=2^a\text{ for a positive integer }a.$$

Suppose that a profinite group $G$ satisfies $P_2(G)>0$, where $P_2(G)$ denotes the proportion of 2-elements in the relevant finite quotients. Exceptional simple-factor conjecture. Either $G$ is virtually prosolvable or, for every $k\in\mathbb N$, there exists an open normal subgroup $N$ of $G$ such that a composition series of $G/N$ has at least $k$ factors belonging to $\Sigma$.

This conjecture describes the only proposed obstruction to virtual prosolvability when a profinite group has a positive proportion of 2-elements: arbitrarily many composition factors from the exceptional family may occur in finite quotients. The preceding results establish the relevant exceptional behavior for the groups in $\Sigma$, but the proposed dichotomy remains unresolved.